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The question of the maximum number $\mbox{ex}(m,n,C_{2k})$ of edges in an m by n bipartite graph without a cycle of length 2k is addressed in this note. For each $k \geq 2$, it is shown that $\mbox{ex}(m,n,C_{2k}) \leq \begin{cases} (2k-3)\bigl[(mn)^{\frac{k+1}{2k}} + m + n\bigr] & \mbox{ if }k \mbox{ is odd,}\\[2pt] (2k-3)\bigl[m^{\frac{k+2}{2k}}\, n^{\frac{1}{2}} + m + n\bigr] & \mbox{ if }k \mbox{ is even.}\\ \end{cases}$